Question

eh bisects ∠feg and (overline{ef}congoverline{eg}). complete the proof that (∠ehgcong∠ehf).

Explanation:

Step1: Given

Statement 1: \( \overrightarrow{EH}\) bisects \( \angle FEG\). This is given in the problem statement.

Step2: Given

Statement 2: \( \overline{EF}\cong\overline{EG}\). This is also given in the problem statement.

Step3: Definition of angle - bisector

Since \( \overrightarrow{EH}\) bisects \( \angle FEG\), by the definition of an angle - bisector, \( \angle FEH\cong\angle GEH\).

Step4: Reflexive property of congruence

For any segment \( \overline{AB}\), \( \overline{AB}\cong\overline{AB}\). So, \( \overline{EH}\cong\overline{EH}\) (reflexive property).

Step5: SAS (Side - Angle - Side) congruence criterion

We have \( \overline{EF}\cong\overline{EG}\) (side), \( \angle FEH\cong\angle GEH\) (angle), and \( \overline{EH}\cong\overline{EH}\) (side). So, \( \triangle EFH\cong\triangle EGH\) by the SAS congruence criterion.

Step6: Corresponding parts of congruent triangles are congruent (CPCTC)

Since \( \triangle EFH\cong\triangle EGH\), their corresponding angles \( \angle EHG\) and \( \angle EHF\) are congruent. That is, \( \angle EHG\cong\angle EHF\) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of angle - bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)